Extended Barrier Transport with Leakage

T98 extends the coherent-crossing transport pipeline of T97 into a complete two-branch barrier architecture. Rather than tracking only the gate-admitted component, the theorem follows both the admitted and rejected portions of the reduced crossing state, yielding the minimal extended transport model for barrier interaction. Beginning with an incoming fibre state \[ψfull5×5,\] The native-side transport and reduction operators produce the reduced crossing state \[ψPQ=R\,D_NM_N\,ψfull².\] Using the gate-admission projector \(Π_Y\) and leakage complement \[K=I-Π_Y,\] Introduced in T96, the reduced state decomposes orthogonally into \[ψPQ=Π_YψPQ+KψPQ.\] The first component satisfies the \(Y\)-gate condition and is eligible for coherent crossing. The second component fails gate admission and enters the leakage branch. The theorem derives the complete extended transport operator \[Tₑₓₜ={pmatrix}D_QM_Q\,Π_Y\,R\,D_NM_N\\[0.4em]K\,R\,D_NM_N{pmatrix},\] which simultaneously produces both outputs of barrier interaction. Applied to an incoming fibre state, the result is \[Tₑₓₜψfull={pmatrix}D_QM_Q\,Π_Y\,R\,D_NM_N\,ψfull\\[0.4em]K\,R\,D_NM_N\,ψfull{pmatrix}.\] The upper component is the coherent crossing branch propagating through the barrier into the dual-side sector. The lower component is the leakage branch consisting of phase content that reaches the reduced crossing sector but fails the \(Y\)-gate condition. A key consequence is that generic barrier interaction is neither complete transmission nor complete rejection. Most states decompose into both branches simultaneously: \[ψPQ=Π_YψPQ+KψPQ.\] The barrier, therefore, acts as a filter rather than a binary gate. Coherent crossing and leakage are complementary outcomes of the same reduced crossing state. The theorem further establishes the complete-crossing and complete-leakage limits. A state crosses cleanly if and only if \[K\,R\,D_NM_N\,ψfull=0,\] equivalently, \[R\,D_NM_N\,ψfull\{|Y\}.\] Conversely, a state exhibits complete leakage if and only if \[Π_Y\,R\,D_NM_N\,ψfull=0,\] so that no coherent crossing component survives. Using the orthogonal decomposition established in T96, the theorem preserves the norm split \[C_Y+D_Y=1,\] where \[C_Y=\|Π_YψPQ\|^2\] is the gate-admission weight and \[D_Y=\|KψPQ\|^2\] is the leakage weight. A major consequence of the theorem is that the leakage branch becomes the explicit source of the residual return dynamics developed in T94. Substituting the reduced crossing state into the return formula yields \[R(t)=K^ etBK\,R\,D_NM_N\,ψfull,\] providing a complete operator pathway from incoming fibre state to leakage circulation and subsequent return. T98 therefore completes the transport architecture initiated in T97. The barrier is revealed as a two-output filter that separates an incoming reduced state into an admitted crossing component and a rejected leakage component. Together with T93–T97, the theorem provides the first complete operator-level description of barrier transport, leakage generation, and residual return within the Q5 framework.